# How a self-attention layer can learn convolutional filters?

Self-attention had a great impact on text processing and became the de-facto building block for NLUNatural Language Understanding. But this success is not restricted to text (or 1D sequences)—transformer-based architectures can beat state of the art ResNets on vision tasks. In an attempt to explain this achievement, our work shows that self-attention can express a CNN layer and that convolutional filters are learned in practice. We provide an interactive website to explore our results.

The transformer architecture introduced by Ashish Vaswani and colleagues has become the workhorse of Natural Language Understanding. The key difference between transformers and previous methods, such as recurrent neural networks (RNN) and convolutional neural networks (CNN), is that transformers can simultaneously attend to every word of their input sequence.

Recently, researchers at Google AI successfully applied the transformer architecture to images (instead of text). This implied replacing all CNN layers by self-attention and adjusting the number of parameters for a fair comparison. This article is an introduction to the paper On the Relationship between Self-Attention and Convolutional Layers (under review), in which we investigate how transformers process images. Specifically, we show that a multi-head self-attention layer with sufficient number of heads can be at least as expressive as any convolutional layer. Our finding presents a possible explanation for the success of transformers on images.

## Self-Attention & Convolutional Layers

To point out the similarities and the differences between a convolutional layer and a self-attention layer, we first recall how each of them process an image of shape $$W \times H \times D_{\textit{in}}$$. The mechanisms behind CNNs are very well understood, however lifting the transformer architecture from 1D (text) to 2D (image) necessitates having a good grasp of self-attention mechanics. You can refer to Attention is All You Need paper and The Illustrated Transformer blog post for explaination of the 1D case.

### Convolutional Layer

A Convolutional Neural Netword (CNN) is composed of many convolutional layers and subsampling layers. Each convolutional layer learns convolutional filters of size $$K \times K$$, with input and output dimensions $$D_{in}$$ and $$D_{out}$$, respectively. The layer is parametrized by a 4D kernel tensor $$\mathbf{W}$$ of dimension $$K \times K \times D_{in} \times D_{out}$$ and a bias vector $$\mathbf{b}$$ of dimension $$D_{out}$$.

The following figure depicts how the output value of a pixel $$\mathbf{q}$$ is computed. In the animation, we consider each shift $$\Delta\!\!\!\!\Delta = [-\lfloor K / 2 \rfloor, ..., \lfloor K / 2 \rfloor]^2$$ of the kernel separately. This view might be unconventional, but it will prove helpful in the following when we will compare convolutional and self-attention layers.

The main difference between CNN and self-attention layers is that the new value of a pixel depends on every other pixel of the image. As opposed to convolution layers whose receptive field is the $$K\times K$$ neighborhood grid, the self-attention's receptive field is always the full image. This brings some scaling challenges when we apply transformers to images that we don't cover here. For now, let's define what is a multi-head self-attention layer.

A self attention layer is defined by a key/query size $D_k$, a head size $$D_h$$, a number of heads $N_h$ and an output dimenson $$D_{\textit{out}}$$. The layer is parametrized by a key matrix $$\mathbf{W}^{(h)}_{\!\textit{key}}$$, a query matrix $$\mathbf{W}^{(h)}_{\!\textit{qry}}$$ and a value matrix $$\mathbf{W}^{(h)}_{\!\textit{val}}$$ for each head $$h$$, along with a projection matrix $$\mathbf{W}_{\!\textit{out}}$$ used to assemble all heads together.

The computation of the attention probabilities is based on the input values $$\mathbf{X}$$. This tensor is often augmented (by addition or concatenation) with positional encodings to distinquish between pixel positions in the image. The hypothetical examples of attention proabilities patterns illustrate dependencies on pixel values and/or positions: uses the values of the query and key pixels, only uses the key pixel positional encoding, uses both the value of the key pixels and their positions.

### Reparametrization

You might already see the similarity between self-attention and convolutional layers. Let's assume that each pair of key/query matrices, $$\mathbf{W}^{(h)}_{\!\textit{key}}$$ and $$\mathbf{W}^{(h)}_{\!\textit{qry}}$$ can attend specifically to a single pixel at any shift $\mathbf{\Delta}$ (producing an attention probability map similar to Figure 1). Then, each attention head would learn a value matrix $$\mathbf{W}^{(h)}_{\!\textit{val}}$$ analogous to the convolutional kernel $$\mathbf{W}_{\mathbf{\Delta}}$$ (both in green on the figures) for each shift $$\mathbf{\Delta}$$. Hence, the number of pixels in the receptive field of the convolutional kernel is related to the number of heads by $$N_h = K \times K$$. This intuition is stated more formally in the following theorem (proved in our paper).

#### Theorem

A multi-head self-attention layer with N_h heads of dimension D_h, output dimension D_{\textit{out}} and a relative positional encoding of dimension D_p \geq 3 can express any convolutional layer of kernel size \sqrt{N_h} \times \sqrt{N_h} and \text{min}(D_h, D_{\textit{out}}) output channels.

The two most crucial requirements for a self-attention layer to express a convolution are:

• having multiple heads to attend to every pixel of a convolutional layer's receptive field,
• using relative positional encoding to ensure translation equivariance.

The first point might give the first explanation why multi-head attention works better than single-head. Regarding the second point, we next give insights on how to encode positions to ensure that self-attention can compute a convolution.

## Relative Positional Encoding

A key property of the self-attention model described above is that it is equivariant to reordering, that is, it gives the same output independently of how the input pixels are shuffled. This is problematic for cases we expect the order of input to matter. To alleviate the limitation, a positional encoding is learned for each token in the sequence (or pixel in an image), and added to the representation of the token itself before applying self-attention.

The attention probabilites (Figure 3, top right) are computed based on the input values and the positional encoding of the layer input. We have already seen that each head can focus on different part (position or content) of the image for each query pixel. We can explicitly decompose these different dependencies as follows:

Because the receptive field of a convolution layer does not depend on the input data, only the last term is needed for the self-attention to emulate a CNN.

An important property of CNN that we are missing is equivariance to translation. This can be achieved by replacing the absolute positional encoding by relative positional encoding $$\mathbf{r}_\mathbf{\delta}$$. This encoding was first introduced by Zihang Dai and colleagues in TransformerXL. The main idea is to only consider the position difference $$\mathbf{\delta} = \mathbf{k} - \mathbf{q}$$ between the key pixel (pixel we attend) and the query pixel (pixel we compute the representation of) instead of the absolute position of the key pixel. The absolution attention probabilities can then be rewritten in a relative manner (refer to the paper for the new matrices and vectors parameters):

In this manner, the attention scores only depend on the shift and we achieve translation equivariance. Finally we show that there exists a set of relative positional vectors of dimension $$D_{\textit{pos}} = 3$$ along with self-attention parameters that allow attending to pixels at arbitrary shift (Figure 1). We conclude that any convolutional filter can be expressed by a multi-head self-attention under conditions stated in the theorem above.

## Learned Attention Patterns

Even though we proved that self-attention layers have the capacity to express any convolutional layer, this does not necessarily mean that the behavior occurs in practice. To verify our hypothesis, we implemented a fully-attentional model of 6 layers, with 9 heads each. We trained a supervised classification objective on CIFAR-10 and reached a good accuracy of 94% (not state of the art, but good enough). We re-used the learned relative positional encoding from Irawn Bello and colleagues, learning separatly row and column offset encoding. The main difference is that we only used the relative positions to condition the attention probabilities, not the input values.

The attention probabilities displayed on Figure 4 show that, indeed, self-attention behaves similarly to convolution. Each head learns to focus on different parts of the images, important attention probabilities are in general very localized.

We can also observe that the first layers (1-3) concentrate on very close and specific pixels while deeper layers (4-6) are attending on more global patches of pixels over whole regions of the image. In the paper, we further experimented with more heads and obserded more complex (learned) patterns than a grid of pixels.

### Conclusion

In our paper On the Relationship between self-attention and Convolutional Layers, we showed that a self-attention layer can express any convolutional filters given enough heads and using relative positional encoding. This is maybe the first explanation why multiple heads are necessary. In fact, the multi-head self-attention layer generalizes the convolutional layer: it learns the positions of its receptive field on the whole image (instead of a fixed grid). The recepteive field can even be conditioned on the value of the input pixels, we left this interesting feature for future work. Hopefully, our findings on positional encoding for images can also be usefull for text, it seems that using learned relative positional encoding of dimension 2 should be enough but we need to verify this in practice.

### Acknowledgments

To lighten this article and point the reader only to the most relevant papers, we cited only a subset of the relevant work that we built on. Please refer to the bibliography of the original paper for the complete list.

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